ISSN: 1525-3066
Center for Policy Research
Working Paper No. 75
ON THE ESTIMATION AND INFERENCE
OF A PANEL COINTEGRATION MODEL
WITH CROSS-SECTIONAL DEPENDENCE
Jushan Bai and Chihwa Kao
Center for Policy Research
Maxwell School of Citizenship and Public Affairs
Syracuse University
426 Eggers Hall
Syracuse, New York 13244-1020
(315) 443-3114 | Fax (315) 443-1081
e-mail: [email protected]
December 2005
$5.00
Up-to-date information about CPR’s research projects and other activities is
available from our World Wide Web site at www-cpr.maxwell.syr.edu. All
recent working papers and Policy Briefs can be read and/or printed from there as
well.
CENTER FOR POLICY RESEARCH – Fall 2005
Timothy Smeeding, Director
Professor of Economics & Public Administration
__________
Associate Directors
Margaret Austin
Associate Director,
Budget and Administration
Douglas Holtz-Eakin
Professor of Economics
Associate Director, Center for Policy Research
Douglas Wolf
Professor of Public Administration
Associate Director, Aging Studies Program
John Yinger
Professor of Economics and Public Administration
Associate Director, Metropolitan Studies Program
SENIOR RESEARCH ASSOCIATES
Badi Baltagi............................................ Economics
Pablo Beramendi............................Political Science
Dan Black............................................... Economics
Lloyd Blanchard ..................... Public Administration
William Duncombe ................. Public Administration
Gary Engelhardt ....................................Economics
Deborah Freund .................... Public Administration
Madonna Harrington Meyer .....................Sociology
Christine Himes........................................Sociology
William C. Horrace .................................Economics
Bernard Jump ........................Public Administration
Duke Kao ...............................................Economics
Eric Kingson ........................................ Social Work
Thomas Kniesner ..................................Economics
Tom Krebs ............................................. Economics
Jeff Kubik ...............................................Economics
Andrew London ......................................... Sociology
Len Lopoo ............................... Public Administration
Jerry Miner .............................................. Economics
John Moran ............................................. Economics
Jan Ondrich............................................. Economics
John Palmer ............................ Public Administration
Lori Ploutz-Snyder.... Health and Physical Education
Grant Reeher ..................................Political Science
Stuart Rosenthal ..................................... Economics
Ross Rubenstein..................... Public Administration
Margaret Usdansky…………………………Sociology
Michael Wasylenko ................................. Economics
Janet Wilmoth ........................................... Sociology
GRADUATE ASSOCIATES
Javier Baez ............................................Economics
Sonali Ballal………………….. Public Administration
Dana Balter ............................Public Administration
Jason Bladen .........................................Economics
Jesse Bricker .........................................Economics
Maria Brown .....................................Social Science
Yong Chen .............................................Economics
Ginnie Cronin ...................................Social Science
Ana Dammert .........................................Economics
Mike Eriksen ..........................................Economics
Katie Fitzpatrick......................................Economics
Alexandre Genest .................. Public Administration
Julie Anna Golebiewski ..........................Economics
Nadia Greenhalgh-Stanley ....................Economics
Yue Hu .................................................... Economics
Becky Lafrancois..................................... Economics
Joseph Marchand.................................... Economics
Larry Miller .............................. Public Administration
Wendy Parker ........................................... Sociology
Emily Pas ................................................ Economics
Shawn Rohlin .......................................... Economics
Cynthia Searcy........................ Public Administration
Talisha Searcy ........................ Public Administration
Jeff Thompson ........................................ Economics
Wen Wang .............................................. Economics
Yoshikun Yamamoto ............... Public Administration
STAFF
Kelly Bogart ......................Administrative Secretary
Martha Bonney..... Publications/Events Coordinator
Karen Cimilluca ........... Librarian/Office Coordinator
Kim Desmond …………….Administrative Secretary
Kati Foley……………...Administrative Assistant, LIS
Kitty Nasto……………….…Administrative Secretary
Candi Patterson.......................Computer Consultant
Mary Santy……...….………Administrative Secretary
On the Estimation and Inference of a Panel
Cointegration Model with Cross-Sectional Dependence
Jushan Bai
Department of Economics
New York University
New York, NY 10003
USA
and
Department of Economics
Tsinghua University
Beijing 10084
China
[email protected]
Chihwa Kao
Center for Policy Research
and
Department of Economics
Syracuse University
Syracuse, NY 13244-1020
USA
[email protected]
First Draft: March 2003
This Draft: May 2, 2005
Abstract
Most of the existing literature on panel data cointegration assumes crosssectional independence, an assumption that is difficult to satisfy. This paper
studies panel cointegration under cross-sectional dependence, which is characterized by a factor structure. We derive the limiting distribution of a fully modified
estimator for the panel cointegrating coefficients. We also propose a continuousupdated fully modified (CUP-FM) estimator. Monte Carlo results show that the
CUP-FM estimator has better small sample properties than the two-step FM
(2S-FM) and OLS estimators.
1
1
Introduction
A convenient but difficult to justify assumption in panel cointegration analysis is crosssectional independence. Left untreated, cross-sectional dependence causes bias and inconsistency estimation, as argued by Andrews (2003). In this paper, we use a factor structure
to characterize cross-sectional dependence. Factors models are especially suited for this
purpose. One major source of cross-section correlation in macroeconomic data is common
shocks, e.g., oil price shocks and international financial crises. Common shocks drive the
underlying comovement of economic variables. Factor models provide an effective way to
extract the comovement and have been used in various studies.1 Cross-sectional correlation
exists even in micro level data because of herd behavior (fashions, fads, and imitation cascades) either at firm level or household level. The general state of an economy (recessions
or booms) also affects household decision making. Factor models accommodate individual’s
different responses to common shocks through heterogeneous factor loadings.
Panel data models with correlated cross-sectional units are important due to increasing
availability of large panel data sets and increasing interconnectedness of the economies.
Despite the immense interest in testing for panel unit roots and cointegration,2 not much
attention has been paid to the issues of cross-sectional dependence. Studies using factor
models for nonstationary data include Bai and Ng (2004), Bai (2004), Phillips and Sul (2003),
and Moon and Perron (2004). Chang (2002) proposed to use a nonlinear IV estimation to
construct a new panel unit root test. Hall et al (1999) considered a problem of determining
the number of common trends. Baltagi et al. (2004) derived several Lagrange Multiplier tests
for the panel data regression model with spatial error correlation. Robertson and Symons
(2000), Coakley et al. (2002) and Pesaran (2004) proposed to use common factors to capture
the cross-sectional dependence in stationary panel models. All these studies focus on either
stationary data or panel unit root studies rather than panel cointegration.
This paper makes three contributions. First, it adds to the literature by suggesting a
factor model for panel cointegrations. Second, it proposes a continuous-updated fully modified (CUP-FM) estimator. Third, it provides a comparison for the finite sample properties
1
For example, Stock and Watson (2002), Gregory and Head (1999), Forni and Reichlin (1998) and Forni
et al. (2000) to name a few.
2
see Baltagi and Kao (2000) for a recent survey
of the OLS, two-step fully modified (2S-FM), CUP-FM estimators.
The rest of the paper is organized as follows. Section 2 introduces the model. Section
3 presents assumptions. Sections 4 and 5 develop the asymptotic theory for the OLS and
fully modified (FM) estimators. Section 6 discusses a feasible FM estimator and suggests
a CUP-FM estimator. Section 7 makes some remarks on hypothesis testing. Section 8
presents Monte Carlo results to illustrate the finite sample properties of the OLS and FM
estimators. Section 9 summarizes the findings. The appendix contains the proofs of Lemmas
and Theorems.
R
The following notations are used in the paper. We write the integral
R1
0
W (s)ds as
W when there is no ambiguity over limits. We define Ω1/2 to be any matrix such that
¡
¢¡
¢0
© ¡ 0 ¢ª1/2
Ω = Ω1/2 Ω1/2 . We use kAk to denote tr A A
, |A| to denote the determinant of
p
A, ⇒ to denote weak convergence, → to denote convergence in probability, [x] to denote
the largest integer ≤ x, I(0) and I(1) to signify a time-series that is integrated of order zero
and one, respectively, and BM (Ω) to denote Brownian motion with the covariance matrix
Ω. We let M < ∞ be a generic positive number, not depending on T or n.
2
The Model
Consider the following fixed effect panel regression:
yit = αi + βxit + eit , i = 1, ..., n, t = 1, ..., T,
(1)
where yit is 1 × 1, β is a 1 × k vector of the slope parameters, αi is the intercept, and eit is
the stationary regression error. We assume that xit is a k × 1 integrated processes of order
one for all i, where
xit = xit−1 + εit .
Under these specifications, (1) describes a system of cointegrated regressions, i.e., yit is
cointegrated with xit . The initialization of this system is yi0 = xi0 = Op (1) as T → ∞ for all
i. The individual constant term αi can be extended into general deterministic time trends
such as α0i + α1i t+, ..., +αpi t or other deterministic component. To model the cross-sectional
dependence we assume the error term, eit , follows a factor model (e.g., Bai and Ng, 2002,
2004):
0
eit = λi Ft + uit ,
2
(2)
where Ft is a r × 1 vector of common factors, λi is a r × 1 vector of factor loadings and uit
is the idiosyncratic component of eit , which means
³
´
0
0
E (eit ejt ) = λi E Ft Ft λj
i.e., eit and ejt are correlated due to the common factors Ft .
Remark 1
1. We could also allow εit to have a factor structure such that
0
εit = γ i Ft + ηit .
Then we can use 4xit to estimate Ft and γ i . Or we can use eit together with 4xit to
estimate Ft , λi and γ i . In general, εit can be of the form
εit = γ 0i Ft + τ 0i Gt + ηit ,
where Ft and Gt are zero mean processes, and ηit are usually independent over i and
t.
3
Assumptions
Our analysis is based on the following assumptions.
Assumption 1 As n → ∞,
1
n
Pn
i=1
0
λi λi −→ Σλ , a r × r positive definite matrix.
¡ 0
P∞
0 ¢0
Assumption 2 Let wit = Ft , uit , εit . For each i, wit = Πi (L)vit =
j=0 Πij vit−j ,
P∞ a
j=0 j kΠij k < ∞, |Πi (1)| 6= 0 for some a > 1, where vit is i.i.d. over t. In addition,
Evit = 0, E(vit vit0 ) = Σv > 0, and Ekvit k8 ≤ M < ∞.
Assumption 3 Ft and uit are independent; uit are independent across i.
Under Assumption 2, a multivariate invariance principle for wit holds, i.e., the partial
P r]
sum process √1T [T
t=1 wit satisfies:
[T r]
1 X
√
wit ⇒ B (Ωi ) as T → ∞ for all i,
T t=1
3
(3)
where
⎡
⎤
BF
⎢
⎥
Bi = ⎣ Bui ⎦ .
Bεi
The long-run covariance matrix of {wit } is given by
Ωi =
∞
X
j=−∞
³
´
0
E wi0 wij
0
= Πi (1)Σv Πi (1)
0
= Σi + Γi + Γi
⎤
⎡
ΩF i ΩF ui ΩF εi
⎥
⎢
= ⎣ ΩuF i Ωui Ωuεi ⎦ ,
ΩεF i Ωεui Ωεi
where
Γi =
∞
X
j=1
and
⎡
ΓF i
ΓF ui ΓF εi
³
´
0
⎢
E wi0 wij = ⎣ ΓuF i
Γui
ΓεF i
⎡
ΣF i
³
´
0
⎢
Σi = E wi0 wi0 = ⎣ ΣuF i
ΣεF i
Γεui
⎥
Γuεi ⎦
Γεi
ΣF ui ΣF εi
Σui
Σεui
are partitioned conformably with wit . We denote
⎤
⎤
⎥
Σuεi ⎦
Σεi
n
1X
Ω = lim
Ωi ,
n→∞ n
i=1
n
and
1X
Γ = lim
Γi ,
n→∞ n
i=1
n
1X
Σ = lim
Σi .
n→∞ n
i=1
Assumption 4 Ωεi is non-singular, i.e., {xit } are not cointegrated.
4
(4)
Define
Ωbi =
"
ΩF i
ΩF ui
ΩuF i
Ωui
#
,
Ωbεi =
"
ΩF εi
Ωuεi
#
and
Ωb.εi = Ωbi − Ωbεi Ω−1
εi Ωεbi .
Then, Bi can be rewritten as
"
Bi =
Bbi
Bεi
#
=
"
1/2
Bbi =
Vbi =
"
Vbi
Wi
1/2
0
where
and
−1/2
Ωb.εi Ωbεi Ωεi
#
Ωεi
"
"
BF
Bui
VF
Vui
#
#
#"
Vbi
Wi
#
,
(5)
,
,
= BM (I)
is a standardized Brownian motion. Define the one-sided long-run covariance
∆i = Σi + Γi
∞
³
´
X
0
=
E wi0 wij
j=0
with
∆i =
Remark 2
"
∆bi
∆bεi
∆εbi ∆εi
#
.
1. Assumption 1 is a standard assumption in factor models (e.g., Bai and Ng
2002, 2004) to ensure the factor structure is identifiable. We only consider nonrandom
0
factor loadings for simplicity. Our results still hold when the λi s are random, provided
they are independent of the factors and idiosyncratic errors, and E kλi k4 ≤ M.
¡
0 ¢
2. Assumption 2 assumes that the random factors, Ft , and idiosyncratic shocks uit , εit
are stationary linear processes. Note that Ft and εit are allowed to be correlated. In
particular, εit may have a factor structure as in Remark 1.
5
3. Assumption of independence made in Assumption 3 between Ft and uit can be relaxed
following Bai and Ng (2002). Nevertheless, independence is not a restricted assumption
since cross-sectional correlations in the regression errors eit are taken into account by
the common factors.
4
OLS
Let us first study the limiting distribution of the OLS estimator for equation (1). The OLS
estimator of β is
bOLS =
β
" n T
XX
i=1 t=1
0
yit (xit − xi )
#" n T
XX
i=1 t=1
Theorem
1 Under´Assumptions 1 −
³ 4, we have
n
√ ³b
√
nT β OLS − β − nδ nT ⇒ N 0, 6Ω−1 lim
1
n→∞ n
ε
n
T
0
(xit − xi ) (xit − xi )
Pn ³
i=1
0
#−1
(6)
.
λi ΩF.εi λi Ωεi + Ωu.εi Ωεi
´o
´
Ω−1
,
ε
as (n, T → ∞) with → 0 where
" n
#
µ
µZ
¶
¶
µZ
¶
1 X 0
0
0
1/2
1/2
fi dWi Ω−1/2
fi dWi Ω−1/2
λ ΩF εi Ωεi
W
+ ∆F εi + Ωuεi Ωεi
W
+ ∆uεi
δ nT =
εi
εi
n i=1 i
" n
#−1
T
0
1X 1 X
(xit − xit ) (xit − xi )
,
n i=1 T 2 t=1
fi = Wi −
W
R
Wi and Ωε = lim n1
n→∞
Pn
i=1
Ωεi .
Remark 3 It is also possible to construct the bias-corrected OLS by using the averages of
the long run covariances. Note
" n
#µ
µ
¶
¶−1
1 X 0
1
1
1
E [δ nT ] '
λ − ΩF εi + ∆F εi − Ωuεi + ∆uεi
Ωε
n i=1 i
2
2
6
" n µ ¶
#µ
¶−1
´
1 ³ 0
1 X
1
0
=
−
λi ΩF εi + Ωuεi + λi ∆F εi + ∆uεi
Ωε
n i=1
2
6
à n µ ¶
!µ
¶−1
n
n
n
1X
1
1X
1X 0
1X
1
0
=
−
λi ΩF εi +
Ωuεi +
λ ∆F εi +
∆uεi
Ωε
n i=1
2
n i=1
n i=1 i
n i=1
6
6
It can be shown by a central limit theorem that
√
n (δ nT − E [δ nT ]) ⇒ N (0, B)
for some B. Therefore,
√
nT
√
nT
=
³
´
bOLS − β − √nE [δ nT ]
β
³
´ √
√
b
β OLS − β − nδ nT + n (δ nT − E [δ nT ])
⇒ N (0, A)
for some A.
5
FM Estimator
bF M . The FM estimator
Next we examine the limiting distribution of the FM estimator, β
was suggested by Phillips and Hansen (1990) in a different context (non-panel data). The
FM estimator is constructed by making corrections for endogeneity and serial correlation to
bOLS in (6). The endogeneity correction is achieved by modifying the
the OLS estimator β
variable yit in (1) with the transformation
³ 0
´
yit+ = yit − λi ΩF εi + Ωuεi Ω−1
εi ∆xit .
The serial correlation correction term has the form
−1
∆+
bεi = ∆bεi − Ωbεi Ωεi ∆εi ,
#
"
∆+
F εi
.
=
∆+
uεi
Therefore, the infeasible FM estimator is
eF M =
β
" n à T
X X
i=1
t=1
!# " n T
#−1
³ 0
´
XX
0
+
yit+ (xit − xi ) − T λi ∆+
(xit − xi ) (xit − xi )
.
F εi + ∆uεi
0
i=1 t=1
(7)
eF M .
Now, we state the limiting distribution of β
7
Theorem 2 Let Assumptions 1 − 4 hold. Then as (n, T → ∞) with Tn → 0
Ã
(
)
!
n ³
´
´
X
√ ³e
1
0
nT β F M − β ⇒ N 0, 6Ω−1
lim
.
λi ΩF.εi λi Ωεi + Ωu.εi Ωεi
Ω−1
ε
ε
n→∞ n
i=1
Remark 4 The asymptotic distribution in Theorem 2 is reduced to
ÃÃ
!
!!
Ã
n
´
X
√ ³e
1
lim
nT β F M − β ⇒ N 0, 6Ω−1
λ2i ΩF.ε + Ωu.ε
ε
n→∞ n
i=1
if the long-run covariances are the same across the cross-sectional unit i and r = 1.
6
Feasible FM
In this section we investigate the limiting distribution of the feasible FM. We will show that
the limiting distribution of the feasible FM is not affected when λi , Ωεi , Ωεbi , Ωεi , and ∆εbi
are estimated. To estimate λi , we use the method of principal components used in Stock and
0
0
Watson (2002). Let λ = (λ1 , λ2 , ..., λn ) and F = (F1 , F2 , ..., FT ) . The method of principal
components of λ and F minimizes
T
n
´2
1 XX³
0
V (r) =
eit − λi Ft
b
nT i=1 t=1
where
b it
yit − α
b i − βx
b (xit − xi ) ,
= (yit − y i ) − β
eit =
b
b
with a consistent estimator β.
Concentrating out λ and using the normalization that
¡ 0 ¡ 0¢ ¢
F F/T = Ir , the optimization problem is identical to maximizing tr F ZZ F , where
0
0
Z = (b
e1 , eb2 , ..., b
en ) is T × n with b
ei = (b
ei1 , ebi2 , ..., b
eiT ) . The estimated factor matrix, denoted
√
by Fb, a T × r matrix, is T times eigenvectors corresponding to the r largest eigenvalues of
0
the T × T matrix ZZ , and
b
λ
0
=
³
0
Fb Fb
0
Fb Z
=
T
8
´−1
0
Fb Z
is the corresponding matrix of the estimated factor loadings. It is known that the solution
to the above minimization problem is not unique, i.e., λi and Ft are not directly identifiable
but they are identifiable up to a transformation H. For our setup, knowing Hλi is as good
0
as knowing λi . For example in (7) using λi ∆+
F εi will give the same information as using
0
0
0
0
0
0
+
−1 +
∆F εi =
λi H H −1 ∆+
F εi since ∆F εi is also identifiable up to a transformation, i.e., λi H H
0
λi ∆+
F εi . Therefore, when assessing the properties of the estimates we only need to consider
bi and λi .
the differences in the space spanned by, say, between λ
bi , Fbt , Σ
bF M , with λ
b i , and Ω
b i in place of λi , Ft , Σi , and Ωi ,
Define the feasible FM, β
b
β
FM =
where
" n à T
X X
i=1
t=1
!# " n T
#−1
³ 0
´
XX
0
b∆
b+
b+
ybit+ (xit − xi ) − T λ
(xit − xi ) (xit − xi )
,
i F εi + ∆uεi
0
i=1 t=1
´
³ 0
b −1 ∆xit .
b F εi + Ω
b uεi Ω
ybit+ = yit − b
λi Ω
εi
b + are defined similarly.
b + and ∆
and ∆
uεi
F εi
Assume that Ωi = Ω for all i. Let
³ 0
´
e+
=
e
−
λ
Ω
+
Ω
Ω−1
it
F
ε
uε
i
ε ∆xit ,
it
X
b+ = 1
b+ ,
∆
∆
bεn
n i=1 bεi
n
and
n
∆+
bεn
1X +
=
∆ .
n i=1 bεi
9
Then
´
√ ³b
e
nT β F M − β
FM
(Ã
! Ã T
!)
n
T
X
X
0
0
1 X
+
b+
= √
−
eb+
e+
it (xit − xi ) − T ∆bεn
it (xit − xi ) − T ∆bεn
nT i=1
t=1
t=1
"
#−1
n
T
0
1 XX
(xit − xi ) (xit − xi )
2
nT i=1 t=1
à T
!#
"
n
³
´
¢
0
1 X X¡ +
+
b+
√
e − e+
b
=
it (xit − xi ) − T ∆bεn − ∆bεn
nT i=1 t=1 it
"
#−1
n
T
0
1 XX
(xit − xi ) (xit − xi )
.
nT 2 i=1 t=1
Before we prove Theorem 3 we need the following lemmas.
´
√ ³b+
+
−
∆
Lemma 1 Under Assumptions 1-4 n ∆
bεn
bεn = op (1).
Lemma 1 can be proved similarly by following Phillips and Moon (1999) and Moon and
Perron (2004).
Lemma 2 Suppose Assumptions 1-4 hold. There exists an H with rank r such that as
(n, T → ∞)
(i)
µ
¶
n
°2
1 X°
1
°b
°
°λi − Hλi ° = Op
n i=1
δ2nT
(ii) let ci (i = 1, 2, ..., n) be a sequence of random matrices such that ci = Op (1) and
Pn
1
2
i=1 kci k = Op (1) then
n
´0
1 X ³b
λi − Hλi ci = Op
n i=1
n
where δ nT
n√ √ o
= min
n, T .
µ
1
δ 2nT
¶
Bai and Ng (2002) showed that for a known b
eit that the average squared deviations
bi and Hλi vanish as n and T both tend to infinity and the rate of convergence is
between λ
10
the minimum of n and T . Lemma 2 can be proved similarly by following Bai and Ng (2002)
that parameter estimation uncertainty for β has no impact on the null limit distribution of
bi .
λ
Lemma 3 Under Assumptions 1-4
as (n, T → ∞) and
T
n
¢
0
1 XX¡ +
√
eit − e+
b
it (xit − xi ) = op (1)
nT i=1 t=1
√
n
T
→ 0.
Then we have the following theorem:
√
Theorem 3 Under Assumptions 1-4 and (n, T → ∞) and Tn → 0
´
√ ³
bF M − β
eF M = op (1).
nT β
In the literature, the FM-type estimators usually were computed with a two-step probOLS . Then, one constructs
cedure, by assuming an initial consistent estimate of β, say β
b(1) . The 2S-FM, denoted
b (1) , and loading, λ
estimates of the long-run covariance matrix, Ω
i
(1)
(1)
(1)
b
b
b
β 2S is obtained using Ω and λi :
" n à T
µ 0(1)
¶!#
X X +(1)
(1)
0
+(1)
+(1)
b
b
b
β
yb
(xit − xi ) − T b
λi ∆
+∆
2S =
it
i=1
F εi
t=1
" n T
XX
i=1 t=1
(xit − xi ) (xit − xi )
0
#−1
.
uεi
(8)
In this paper, we propose a CUP-FM estimator. The CUP-FM is constructed by estimating
bF M , Ω
bi
b and λ
parameters and long-run covariance matrix and loading recursively. Thus β
are estimated repeatedly, until convergence is reached. In the Section 8, we find the CUPFM has a superior small sample properties as compared with the 2S-FM, i.e., CUP-FM has
smaller bias than the common 2S-FM estimator. The CUP-FM is defined as
" n à T
!#
´
³ 0³
´
³
´
³
´´
X X ³
0
bCUP =
bCU P (xit − xi ) − T λ
b β
bCU P ∆
bCUP + ∆
bCU P
b+ β
b+ β
β
ybit+ β
i
uεi
F εi
i=1
t=1
" n T
XX
i=1 t=1
(xit − xi ) (xit − xi )
0
#−1
(9)
.
11
Remark 5
1. In this paper, we assume the number of factors, r, is known. Bai and Ng
(2002) showed that the number of factors can be found by minimizing the following:
µ
¶
µ
¶
n+T
nT
IC(k) = log (V (k)) + k
log
.
nT
n+T
³
0
0
bit = Fbt , u
bit , 4 xit
2. Once the estimates of wit , w
(
(10)
0
bi Fbt .
u
bit = ebit − λ
Ω was estimated by
N
, were estimated, we used
T
n X
X
0
b= 1
Σ
w
bit w
bit
nT i=1 t=1
to estimate Σ, where
X
b= 1
Ω
n i=1
´0
T
l
1X
1X
0
w
bit w
bit +
T t=1
T τ =1
τl
T
³
X
t=τ +1
0
0
w
bit w
bit−τ + w
bit−τ w
bit
´
)
,
(11)
b i and
where τ l is a weight function or a kernel. Using Phillips and Moon (1999), Σ
b i can be shown to be consistent for Σi and Ωi .
Ω
7
Hypothesis Testing
We now consider a linear hypothesis that involves the elements of the coefficient vector β. We
show that hypothesis tests constructed using the FM estimator have asymptotic chi-squared
distributions. The null hypothesis has the form:
H0 : Rβ = r,
(12)
where r is a m × 1 known vector and R is a known m × k matrix describing the restrictions.
bF M is
A natural test statistic of the Wald test using β
"
(
)
#−1
n ³ 0
´0
´
³
´
X
1 2³ b
1
−1
bΩ
b
b
b −1 lim
b
b
b
b
b
6Ω
λ
Ω
+
Ω
Ω
Ω
R
β
−
r
.
W = nT Rβ F M − r
λ
u.εi εi
FM
i F.εi i εi
ε
ε
n→∞ n
6
i=1
(13)
12
It is clear that W converges in distribution to a chi-squared random variable with k
degrees of freedom, χ2k , as (n, T → ∞) under the null hypothesis. Hence, we establish the
following theorem:
Theorem 4 If Assumptions 1−4 hold, then under the null hypothesis (12), with (n, T → ∞) ,
W ⇒ χ2k ,
Remark 6
1. One common application of the theorem 6 is the single-coefficient test: one
of the coefficient is zero; β j = β 0 ,
h
i
R = 0 0 ··· 1 0 ··· 0
and r = 0. We can construct a t-statistic
´
√ ³b
nT β jF M − β 0
tj =
sj
where
"
b −1
s2j = 6Ω
ε
(
(14)
#
)
n
´
1 X ³b 0 b b b
b u.εi Ω
b εi
b −1 ,
lim
λi ΩF.εi λi Ωεi + Ω
Ω
ε
n→∞ n
i=1
jj
the jth diagonal element of
"
(
)
#
n ³ 0
´
X
1
bΩ
b bb
b b
b −1
b −1 lim
λ
Ω
6Ω
i F.εi λi Ωεi + Ωu.εi Ωεi
ε
ε
n→∞ n
i=1
It follows that
tj ⇒ N (0, 1) .
(15)
2. General nonlinear parameter restriction such as H0 : h(β) = 0, where h(·) is k ∗ × 1
vector of smooth functions such that
∂h
0
∂β
has full rank k ∗ can be conducted in a similar
fashion as in Theorem 6. Thus, the Wald test has the following form
³
´0
³
´
bF M Vb −1 h β
bF M
Wh = nT 2 h β
h
where
Vbh−1
⎛ ³ 0 ´⎞
´⎞
³
b
b
∂h β F M
∂h β
FM ⎟
−1 ⎜
b
⎠
⎝
=
Vβ ⎝
⎠
0
∂β
∂β
⎛
13
and
It follows that
b −1
Vbβ = 6Ω
ε
(
)
n
´
1 X ³b 0 b b b
b u.εi Ω
b εi
b −1 .
lim
λi ΩF.εi λΩεii + Ω
Ω
ε
n→∞ n
i=1
(16)
Wh ⇒ χ2k∗
as (n, T → ∞) .
8
Monte Carlo Simulations
In this section, we conduct Monte Carlo experiments to assess the finite sample properties of
OLS and FM estimators. The simulations were performed by a Sun SparcServer 1000 and an
Ultra Enterprise 3000. GAUSS 3.2.31 and COINT 2.0 were used to perform the simulations.
Random numbers for error terms, (Ft∗ , u∗it , ε∗it ) were generated by the GAUSS procedure
RNDNS. At each replication, we generated an n(T + 1000) length of random numbers and
then split it into n series so that each series had the same mean and variance. The first
1, 000 observations were discarded for each series. {Ft∗ } , {u∗it } and {ε∗it } were constructed
with Ft∗ = 0, u∗i0 = 0 and ε∗i0 = 0.
To compare the performance of the OLS and FM estimators we conducted Monte Carlo
experiments based on a design which is similar to Kao and Chiang (2000)
yit = αi + βxit + eit
0
eit = λi Ft + uit ,
and
xit = xit−1 + εit
for i = 1, ..., n, t = 1, ..., T, where
⎛
⎞ ⎛ ∗ ⎞ ⎛
⎞⎛ ∗ ⎞
Ft
Ft
0
0
0
Ft−1
⎜
⎟ ⎜ ∗ ⎟ ⎜
⎟⎜ ∗
⎟
⎝ uit ⎠ = ⎝ uit ⎠ + ⎝ 0 0.3 −0.4 ⎠ ⎝ uit−1 ⎠
εit
ε∗it
ε∗it−1
θ31 θ32 0.6
14
(17)
with
⎛
Ft∗
⎞
⎛⎡
0
⎤ ⎡
1
σ 12 σ 13
⎤⎞
⎜ ∗ ⎟ iid ⎜⎢ ⎥ ⎢
⎥⎟
⎝ uit ⎠ ∼ N ⎝⎣ 0 ⎦ , ⎣ σ 21 1 σ 23 ⎦⎠ .
ε∗it
0
σ 31 σ 32 1
For this experiment, we have a single factor (r = 1) and λi are generated from i.i.d.
N (µλ , 1). We let µλ = 0.1. We generated αi from a uniform distribution, U[0, 10], and set
β = 2. From Theorems 1-3 we know that the asymptotic results depend upon variances and
covariances of Ft, uit and εit . Here we set σ 12 = 0. The design in (17) is a good one since the
endogeneity of the system is controlled by only four parameters, θ31 , θ32 , σ 31 and σ 32 . We
choose θ 31 = 0.8, θ32 = 0.4, σ 31 = −0.8 and θ 32 = 0.4.
The estimate of the long-run covariance matrix in (11) was obtained by using the proce-
dure KERNEL in COINT 2.0 with a Bartlett window. The lag truncation number was set
arbitrarily at five. Results with other kernels, such as Parzen and quadratic spectral kernels,
are not reported, because no essential differences were found for most cases.
Next, we recorded the results from our Monte Carlo experiments that examined the
bOLS in (6), (b) the 2S-FM estimator,
finite-sample properties of (a) the OLS estimator, β
b2S , in (8), (c) the two-step naive FM estimator, β
bb , proposed by Kao and Chiang (2000)
β
FM
bCU P , in (9) and (e) the CUP
and Phillips and Moon (1999), (d) the CUP-FM estimator β
bd which is similar to the two-step naive FM except the iteration
naive FM estimator β
FM
goes beyond two steps. The naive FM estimators are obtained assuming the cross-sectional
independence. The maximum number of the iteration for CUP-FM estimators is set to 20.
The results we report are based on 1, 000 replications and are summarized in Tables 1 - 4.
All the FM estimators were obtained by using a Bartlett window of lag length five as in (11).
the´ Monte
and ´standard
³ Table 1´ reports
³
³ b Carlo
´ means
³
³ d deviations
´ (in parentheses) of
bOLS − β , β
b2S − β , β
b
b
b
β
F M − β , β CU P − β ,and β F M − β for sample sizes T = n
bOLS , decrease at a rate of T . For example,
= (20, 40, 60) . The biases of the OLS estimator, β
with σ λ = 1 and σ F = 1, the bias at T = 20 is − 0.045, at T = 40 is −0.024, and at T = 60
is −0.015. Also, the biases stay the same for different values of σ λ and σ F .
TABLE 1 ABOUT HERE
15
While we expected the OLS estimator to be biased, we expected FM estimators to produce
better estimates. However, it is noticeable that the 2S-FM estimator still has a downward
bias for all values of σ λ and σ F , though the biases are smaller. In general, the 2S-FM
estimator presents the same degree of difficulty with bias as does the OLS estimator. This
is probably due to the failure of the nonparametric correction procedure.
In contrast, the results in Table 1 show that the CUP-FM, is distinctly superior to the
OLS and 2S-FM estimators for all cases in terms of the mean biases. Clearly, the CUP-FM
outperforms both the OLS and 2S-FM estimators.
TABLE 2 ABOUT HERE
It is important to know the effects of the variations in panel dimensions on the results,
since the actual panel data have a wide variety of cross-section and time-series dimensions.
Table 2 considers 16 different combinations for n and T , each ranging from 20 to 120 with
σ 31 = −0.8, σ 21 = −0.4, θ31 = 0.8, and θ 21 = 0.4. First, we notice that the cross-section
dimension has no significant effect on the biases of all estimators. From this it seems that in
practice the T dimension must exceed the n dimension, especially for the OLS and 2S-FM
estimators, in order to get a good approximation of the limiting distributions of the estimators. For example, for OLS estimator in Table 2, the reported bias, −0.008, is substantially
less for (T = 120, n = 40) than it is for either (T = 40, n = 40) , (the bias is −0.024), or
(T = 40, n = 120) , (the bias is −0.022). The results in Table 2 again confirm the superiority
of the CUP-FM.
TABLE 3 ABOUT HERE
Monte Carlo means and standard deviations of the t-statistic, tβ=β 0 , are given in Table
3. Here, the OLS t-statistic is the conventional t-statistic as printed by standard statistical
√
packages. With all values of σ λ and σ F with the exception σ λ = 10, the CUP-FM t-statistic
is well approximated by a standard N(0,1) suggested from the asymptotic results. The CUPFM t-statistic is much closer to the standard normal density than the OLS t-statistic and
the 2S-FM t-statistic. The 2S-FM t-statistic is not well approximated by a standard N(0,1).
16
TABLE 4 ABOUT HERE
Table 4 shows that both the OLS t-statistic and the FM t-statistics become more negatively biased as the dimension of cross-section n increases. The heavily negative biases of the
2S-FM t-statistic in Tables 3-4 again indicate the poor performance of the 2S-FM estimator.
For the CUP-FM, the biases decrease rapidly and the standard errors converge to 1.0 as T
increases.
b in (11) may be
It is known that when the length of time series is short the estimate Ω
sensitive to the length of the bandwidth. In Tables 2 and 4, we first investigate the sensitivity
of the FM estimators with respect to the choice of length of the bandwidth. We extend the
experiments by changing the lag length from 5 to other values for a Barlett window. Overall,
the results (not reported here) show that changing the lag length from 5 to other values does
not lead to substantial changes in biases for the FM estimators and their t-statistics.
9
Conclusion
A factor approach to panel models with cross-sectional dependence is useful when both the
time series and cross-sectional dimensions are large. This approach also provides significant
reduction in the number of variables that may cause the cross-sectional dependence in panel
data. In this paper, we study the estimation and inference of a panel cointegration model
with cross-sectional dependence. The paper contributes to the growing literature on panel
data with cross-sectional dependence by (i) discussing limiting distributions for the OLS and
FM estimators, (ii) suggesting a CUP-FM estimator and (iii) investigating the finite sample
proprieties of the OLS, CUP-FM and 2S-FM estimators. It is found that the 2S-FM and
OLS estimators have a non-negligible bias in finite samples, and that the CUP-FM estimator
improves over the other two estimators.
Acknowledgements
We thank Badi Baltagi, Yu-Pin Hu, Giovanni Urga, Kamhon Kan, Chung-Ming Kuan,
Hashem Pesaran, Lorenzo Trapani and Yongcheol Shin for helpful comments. We also thank
17
seminar participants at Academia Sinica, National Taiwan University, Syracuse University,
Workshop on Recent Developments in the Econometrics of Panel Data in London, March
2004 and the European Meeting of the Econometric Society in Madrid, August 2004 for
helpful comments and suggestions. Jushan Bai acknowledges financial support from the
NSF (grant SES-0137084).
Appendix
Let
BnT =
" n T
XX
i=1 t=1
Note
0
(xit − xi ) (xit − xi )
#
.
´
√ ³b
nT β OLS − β
h√ P ³ P
´i £
¤−1
0
1 1
=
n n1 ni=1 T1 Tt=1 eit (xit − xi )
B
n T 2 nT
£√ 1 Pn
¤£ P
¤−1
=
n n i=1 ζ 1iT n1 ni=1 ζ 2iT
√
= nξ 1nT [ξ 2nT ]−1 ,
P
P
P
P
0
0
where xi = T1 Tt=1 xit , y i = T1 Tt=1 yit , ζ 1iT = T1 Tt=1 eit (xit − xi ) , ζ 2iT = T12 Tt=1 (xit − xi ) (xit − xi ) ,
P
P
ξ 1nT = n1 ni=1 ζ 1iT , and ξ 2nT = n1 ni=1 ζ 2iT . Before going into the next theorem, we need to
consider some preliminary results.
P
Define Ωε = lim n1 ni=1 Ωεi and
n→∞
" n
#
µZ
¶
¶
µZ
¶
X 0µ
1
0
0
−1/2
fi dWi Ω1/2 + ∆F εi + Ωuεi Ω−1/2
fi dWi Ω1/2 + ∆uεi .
θn =
λ ΩF.εi Ωεi
W
W
εi
εi
εi
n i=1 i
If Assumptions 1 − 4 hold, then
Lemma 4
(a) As (n, T → ∞) ,
(b) As (n, T → ∞) with
√
n
Ã
n
T
1 1
p 1
BnT → Ωε .
2
nT
6
→ 0,
T
n
0
1 1 XX
eit (xit − xi ) − θn
n T i=1 t=1
!
⇒N
18
Ã
!
n
o
1
1 Xn 0
0, lim
λi ΩF.εi λi Ωεi + Ωu.εi Ωεi
6 n→∞ n i=1
Proof. (a) and (b) can be shown easily by following the Theorem 8 in Phillips and
Moon (1999).
A
Proof of Theorem 1
Proof.
Recall that
⎡ P
³
³
´
´ ⎤
0
0
−1/2 R f
1/2
n
Wi dWi Ωεi + ∆F εi
√
√ 1
i=1 λi ΩF εi Ωεi
b
³
´
⎦
nT β
nn ⎣
OLS − β −
0
−1/2 R f
1/2
+Ωεui Ωεi
Wi dWi Ωεi + ∆εui
£1 1
¤−1
B
n⎡
T 2 nT
⎧
³
³
´
´ ⎫⎤
0
0
−1/2 R f
1/2
⎨
Wi dWi Ωεi + ∆F εi ⎬ £ 1 Pn
ζ 1iT − λi ΩF εi Ωεi
¤−1
√ 1 Pn
⎣
³
´
⎦
=
n n i=1
ζ
R
2iT
i=1
0
−1/2
⎩
fi dWi Ω1/2 + ∆uεi
⎭ n
−Ωuεi Ωεi
W
εi
£√ 1 Pn ∗ ¤ £ 1 Pn
¤−1
=
n n i=1 ζ 1iT n i=1 ζ 2iT
√
= nξ ∗1nT [ξ 2nT ]−1 ,
³
where
ζ ∗1iT
= ζ 1iT
and
´
µ
µZ
¶
¶
µZ
¶
0
0
−1/2
1/2
−1/2
fi dWi Ω + ∆F εi − Ωuε Ω
fi dWi Ω1/2 + ∆uε
− λi ΩF εi Ωεi
W
W
ε
ε
εi
0
n
ξ ∗1nT
1X ∗
=
ζ .
n i=1 1iT
First, we note from Lemma 4 (b) that
!
Ã
n
o
√ ∗
1
1 Xn 0
λi ΩF.εi λi Ωεi + Ωu.εi Ωεi
nξ 1nT ⇒ N 0, lim
6 n→∞ n i=1
as (n, T → ∞) and
√ ∗
nξ 1nT
Hence,
n
T
→ 0. Using the Slutsky theorem and (a) from Lemma 4, we obtain
Ã
(
)
!
n ³
´
X
1
0
[ξ 2nT ]−1 ⇒ N 0, 6Ω−1
lim
λi ΩF.εi λi Ωεi + Ωu.εi Ωεi
Ω−1
.
ε
ε
n→∞ n
i=1
´ √
√ ³b
nT β OLS − β − nδ nT
³
n
´o
´
Pn ³ 0
1
−1
⇒ N 0, 6Ω−1
lim
λ
Ω
λ
Ω
+
Ω
Ω
Ω
,
u.εi εi
ε
ε
i F.εi i εi
i=1
n
n→∞
19
(18)
proving the theorem, where
" n
#
µ
µZ
¶
¶
µZ
¶
1 X 0
0
0
−1/2
1/2
−1/2
1/2
fi dWi Ω + ∆F εi + Ωuεi Ω
fi dWi Ω + ∆uεi
δ nT =
λ ΩF.εi Ωεi
W
W
εi
εi
εi
n i=1 i
∙
¸−1
1 1
BnT
.
n T2
Therefore, we established Theorem 1.
B
Proof of Theorem 2
Proof. Let
Fit+ = Ft − ΩF εi Ω−1
εi εit ,
and
−1
u+
it = uit − Ωuεi Ωεi εit .
The FM estimator of β can be rewritten as follows
h
³P
³ 0
´´i
0
T
+
+
+
eF M = Pn
y
(x
−
x
)
−
T
λ
∆
+
∆
B −1
β
it
i
i=1
hP t=1
³Pit ³ 0
´ i F εi 0 uεi ³ 0 nT
´´i
n
T
+
+
+
+
−1
= β+
λ
F
+
u
(x
−
x
)
−
T
λ
∆
+
∆
BnT
.
it
i
i it
i F εi
it
uεi
i=1
t=1
(19)
³
´
eF M − β by √nT
First, we rescale β
´
´
i£
¤−1
√ ³e
√ 1 Pn 1 PT h³ 0 +
0
0
+
+
+
nT β F M − β =
n n i=1 T t=1 λi Fit + uit (xit − xi ) − λi ∆F εi − ∆uεi n1 T12 BnT
¤−1
£√ 1 Pn ∗∗ ¤ £ 1 Pn
=
n n i=1 ζ 1iT n i=1 ζ 2iT
√ ∗∗
=
nξ 1nT [ξ 2nT ]−1 ,
(20)
h³ 0
´
i
P
P
0
0
T
n
∗∗
∗∗
+
+
1
1
where ζ ∗∗
λi Fit+ + u
b+
1iT = T
it (xit − xi ) − λi ∆F εi − ∆uεi , and ξ 1nT = n
t=1
i=1 ζ 1iT .
Modifying the Theorem 11 in Phillips and Moon (1999) and Kao and Chiang (2000) we
can show that as (n, T → ∞) with
√
n
Ã
n
T
→0
T
n
´
0
1 1 XX³ 0 +
0
+
λ F (xit − xi ) − λi ∆F εi
n T i=1 t=1 i it
20
!
⇒N
Ã
n
1
1X 0
0, lim
λ ΩF.εi λi Ωεi
6 n→∞ n i=1 i
!
and
√
n
Ã
!
Ã
!
T
n
n
´
X
0
1
1
1 1 XX³ +
u
b (xit − xi ) − ∆+
⇒ N 0, lim
Ωu.εi Ωεi
uεi
n T i=1 t=1 it
6 n→∞ n i=1
and combing this with the Assumption 4 that Ft and uit are independent and Lemma 4(a)
yields
´
√ ³e
nT β F M − β ⇒ N
Ã
0, 6Ω−1
ε
(
)
!
n
´
1 X³ 0
lim
λi ΩF.εi λi Ωεi + Ωu.εi Ωεi
Ω−1
ε
n→∞ n
i=1
as required.
C
Proof of Lemma 3
0
bΩ
b F ε is estimating H −10 Ω
b F ε . Thus λ
b
Proof. We note that λi is estimating Hλi , and Ω
i F ε is
0
estimating λi ΩF ε , which is the object of interest. For the purpose of notational simplicity,
we shall assume H being a r × r identify matrix in our proof below. From
³ 0
´
b
b
b
b −1 ∆xit
eb+
=
e
−
λ
Ω
+
Ω
Ω
it
F
ε
uε
i
ε
it
and
e+
it
³ 0
´
= eit − λi ΩF ε + Ωuε Ω−1
ε ∆xit ,
hn³ 0
´
³ 0
´
o
i
−1
−1
bΩ
b
b
b
λ
+
Ω
Ω
−
λ
Ω
+
Ω
Ω
∆x
uε
uε
it
ε
ε
i Fε
i Fε
hn 0
o
i
0
−1
−1
−1
b F εΩ
b −1
b
b
= − b
λi Ω
−
λ
Ω
Ω
+
Ω
Ω
−
Ω
Ω
∆x
.
F
ε
uε
uε
it
ε
ε
ε
ε
i
+
eb+
it − eit = −
Then,
=
T
n
´
0
1 1 X X ³ b b −1
−1
√
Ωuε Ωε − Ωuε Ωε ∆xit (xit − xi )
n T i=1 t=1
³
T
n X
´ 1 1X
0
−1
−1
b
b
Ωuε Ωε − Ωuε Ωε √
∆xit (xit − xi )
n T i=1 t=1
= op (1)Op (1)
= op (1)
21
because
and
Thus
b uε Ω
b −1 − Ωuε Ω−1 = op (1)
Ω
ε
ε
T
n
0
1 1 XX
√
∆xit (xit − xi ) = Op (1).
n T i=1 t=1
T
n
¢
0
1 XX¡ +
√
eit − e+
b
it (xit − xi )
nT i=1 t=1
n
T
´
´
o
i
³ 0
0
1 X 1 X hn³ 0
−1
bΩ
b
b
b
λi ΩF ε + Ωuε Ω−1
Ω
∆x
= √
−
λ
+
Ω
uε
it (xit − xi )
ε
ε
i Fε
n i=1 T t=1
T
n
´
0
0
1 1 XX³ 0
−1
−1
b
b
b
√
λi ΩF ε Ωε − λi ΩF ε Ωε ∆xit (xit − xi )
=
n T i=1 t=1
T
n
´
0
1 1 XX³
−1
−1
b
b
+√
Ωuε Ωε − Ωuε Ωε ∆xit (xit − xi )
n T i=1 t=1
T
n
´
0
0
1 1 XX³ 0
−1
bΩ
b
b
λi ΩF ε Ω−1
∆xit (xit − xi ) + op (1).
= √
−
λ
Ω
F
ε
i
ε
ε
n T i=1 t=1
The remainder of the proof needs to show that
T
n
´
0
0
1 1 XX³ 0
−1
bΩ
b
b
√
λi ΩF ε Ω−1
−
λ
Ω
∆xit (xit − xi ) = op (1) .
ε
i Fε ε
n T i=1 t=1
22
b
b b −1
We write A for ΩF ε Ω−1
ε and A for ΩF ε Ωε respectively and then
T
n
´
0
0
1 1 XX³ 0
−1
bΩ
b
b
√
−
λ
Ω
λi ΩF ε Ω−1
∆xit (xit − xi )
i Fε ε
ε
n T i=1 t=1
T
n
´
0
0
1 1 XX³ 0
bA
b
= √
∆xit (xit − xi )
λi A − λ
i
n T i=1 t=1
T
n
´ ³
i
0´
1 1 XXh 0 ³
b + λ0i − b
b ∆xit (xit − xi )0
= √
λi A − A
λi A
n T i=1 t=1
T
n
´
0
1 1 XX 0 ³
b
√
=
λi A − A ∆xit (xit − xi )
n T i=1 t=1
T
n
0
1 1 X X ³ 0 b0 ´ b
+√
λi − λi A∆xit (xit − xi )
n T i=1 t=1
= I + II, say.
Term I is a row vector. Let Ij be the jth component of I. Let
an identity matrix so that
j
j
0
be the jth column of
= (0, ..., 0, 1, 0, ...0) . Left multiply I by
j
to obtain the jth
component, which is scalar and thus is equal to its trace. That is
Ij
"
³
´
b
A−A
Ã
T
n
0
1 1 XX 0
√
= tr
λi ∆xit (xit − xi )
n T i=1 t=1
i
h
b p (1)
= tr (A − A)O
j λi
!#
= op (1)Op (1)
= op (1)
Pn PT
0
0
b
λ
∆x
(x
−
x
)
it
it
i
j λi = Op (1) and A − A = op (1).
i
i=1
t=1
P
0
b 1 n ∆xit (xit − xi ) . Then ci = Op (1) and
Next consider II. Let ci = A
i=1
T
because
√1 1
nT
23
1
n
Pn
i=1
kci k2 =
Op (1), thus by Lemma 2 (ii), we have
II ≤
=
=
=
=
=
since (n, T → ∞) and
√
n
T
¯
¯
n ³
¯
0´
√ ¯¯ 1 X
0
b ci ¯¯
n¯
λi − λ
i
¯n
¯
i=1
µ
¶
√
1
nOp
2
δ
µ nT
¶
√
1
nOp
min[n, T ]
µ
¶
√
n
Op
min {n, T }
µ
µ√ ¶
¶
1
n
Op √
+ Op
n
T
op (1)
→ 0. This establishes
T
n
´
0
0
1 1 XX³ 0
−1
−1
b
b
b
√
λi ΩF ε Ωε − λi ΩF ε Ωε ∆xit (xit − xi ) = op (1).
n T i=1 t=1
and proves Lemma 3.
References
[1] Andrews, D. W. K., 2003. Cross-section regression with common shocks. forthcoming
in Econometrica.
[2] Bai, J., 2003. Inferential theory for factor models of large dimensions. Econometrica 71,
135-171.
[3] Bai, J., 2004. Estimating cross-section common stochastic trends in nonstationary panel
data. Journal of Econometrics 122, 137-183.
[4] Bai, J., and Ng, S., 2002. Determining the number of factors in approximate factor
models. Econometrica 70, 191-221.
[5] Bai, J., and Ng, S., 2004. A panic attack on unit roots and cointegration. Econometrica
72, 1127-1177.
24
[6] Baltagi, B., and Kao, C., 2000. Nonstationary panels, cointegration in panels and dynamic panels: A survey. Advances in Econometrics 15, 7-51.
[7] Baltagi, B., Song, S. H., and Koh, W., 2004. Testing panel data regression models with
spatial error correlation. Journal of Econometrics 117, 123-150.
[8] Chang, Y., 2002. Nonlinear IV unit root tests in panels with cross-sectional dependency.
Journal of Econometrics 116, 261-292.
[9] Coakley, J., Fuerts, A., and Smith, R. P., 2002. A principal components approach
to cross-section dependence in panels. Manuscript, Birckbeck College, University of
London.
[10] Forni, M. and Reichlin, L., 1998. Let’s get real: a factor-analytic approach to disaggregated business cycle dynamics. Review of Economic Studies 65, 453-473.
[11] Forni, M., Hallin, M., Lippi, M., and Reichlin, L., 2000. Reference cycles: The NBER
methodology revisited. CEPR Discussion Paper 2400.
[12] Gregory, A., and Head, A., 1999. Common and country-specific fluctuations in productivity, investment, and the current account. Journal of Monetary Economics 44,
423-452.
[13] Hall, S. G., Lazarova, S., and Urga, G., 1999. A principle component analysis of common
stochastic trends in heterogeneous panel data: Some monte carlo evidence. Oxford
Bulletin of Economics and Statistics 61, 749-767.
[14] Kao, C., and Chiang, M. H., 2000. On the Estimation and Inference of a Cointegrated
Regression in Panel Data. Advances in Econometrics 15, 179-222.
[15] Moon, H. R., and Perron, B., 2004. Testing for a unit root in panels with dynamic
factors. Journal of Econometrics 122, 81-126.
[16] Pesaran, H., 2004. Estimation and inference in large heterogenous panels with a multifactor error structure. Manuscript, Trinity College, Cambridge.
25
[17] Phillips, P. C. B., and Hansen, B. E., 1990. Statistical inference in instrumental variables
regression with I(1) processes. Review of Economic Studies 57, 99-125.
[18] Phillips, P. C. B., and Moon, H., 1999. Linear regression limit theory for nonstationary
panel data,” Econometrica 67, 1057-1111.
[19] Phillips, P. C. B., and Sul, D., 2003. Dynamic panel estimation and homogeneity testing
under cross section dependence. Econometric Journal 6, 217-259.
[20] Robertson, D., and Symon, J., 2000. Factor residuals in SUR regressions: Estimating
panels allowing for cross sectional correlation. Manuscript, Faculty of Economics and
Politics, University of Cambridge.
[21] Stock, J. H., and Watson, M. W., 2002. Forecasting Using Principal Components from
a Large Number of Predictors. Journal of the American Statistical Association 97 ,
1167—1179.
26
Table 1: Means Biases and Standard Deviation
√ of OLS and FM Estimato
σλ = 1
σ λ = 10
σF = 1
T=20
T=40
T=60
√
σ F = 10
T=20
T=40
T=60
√
σ F = 0.5
T=20
OLS
FMa
FMb
FMc
FMd
OLS
FMa
FMb
FMc
FMd
OLS
F
-0.045
(0.029)
-0.024
(0.010)
-0.015
(0.006)
-0.025
(0.028)
-0.008
(0.010)
-0.004
(0.005)
-0.029
(0.029)
-0.011
(0.010)
-0.005
(0.005)
-0.001
(0.034)
-0.002
(0.010)
-0.001
(0.005)
-0.006
(0.030)
-0.005
(0.010)
-0.003
(0.005)
-0.046
(0.059)
-0.024
(0.020)
-0.015
(0.011)
-0.025
(0.054)
-0.009
(0.019)
-0.003
(0.010)
-0.029
(0.059)
-0.012
(0.019)
-0.005
(0.010)
-0.001
(0.076)
-0.003
(0.021)
-0.001
(0.011)
-0.006
(0.060)
-0.005
(0.018)
-0.002
(0.010)
-0.045
(0.026)
-0.024
(0.009)
-0.015
(0.005)
-0
(0.
-0
(0.
-0
(0.
-0.054
(0.061)
-0.028
(0.021)
-0.018
(0.011)
-0.022
(0.054)
-0.007
(0.019)
-0.002
(0.011)
-0.036
(0.061)
-0.015
(0.019)
-0.007
(0.011)
0.011
(0.078)
0.001
(0.021)
0.001
(0.011)
-0.005
(0.062)
-0.007
(0.019)
-0.004
(0.010)
-0.057
(0.176)
-0.030
(0.059)
-0.017
(0.032)
-0.024
(0.156)
-0.009
(0.054)
-0.001
(0.029)
-0.038
(0.177)
-0.017
(0.057)
-0.006
(0.030)
0.013
(0.228)
-0.001
(0.061)
0.002
(0.031)
-0.003
(0.177)
-0.009
(0.053)
-0.003
(0.029)
-0.054
(0.046)
-0.028
(0.016)
-0.018
(0.009)
-0
(0.
-0
(0.
-0
(0.
-0.002
-0.006
-0.044
(0.056) (0.046) (0.024)
-0.003
-0.005
-0.023
(0.016) (0.014) (0.009)
-0.001
-0.002
-0.015
(0.008) (0.008) (0.005)
the naive CUP-FM
-0
(0.
-0
(0.
-0
(0.
-0.044
(0.026)
T=40
-0.023
(0.009)
T=60
-0.015
(0.005)
Note: (a) FMa is the
-0.025
-0.028
(0.026) (0.026)
-0.009
-0.010
(0.009) (0.009)
-0.004
-0.005
(0.005) (0.005)
2S-FM, FMb is the
-0.003
-0.006
(0.030) (0.028)
-0.003
-0.004
(0.009) (0.009)
-0.001
-0.003
(0.005) (0.005)
naive 2S-FM, FMc
-0.045
-0.026
-0.028
(0.045) (0.041) (0.045)
-0.023
-0.009
-0.011
(0.016) (0.015) (0.015)
-0.015
-0.004
-0.005
(0.009) (0.008) (0.008)
is the CUP-FM and FMd is
(b) µλ = 0.1, σ 31 = −0.8, σ 21 = −0.4, θ31 = 0.8, and θ21 = 0.4.
Table 2: Means Biases and Standard Deviation
of OLS and FM Estimators for Different n and T
(n,T)
OLS
FMa
FMb
FMc
FMd
(20, 20)
-0.045 -0.019 -0.022 -0.001
-0.006
(0.029) (0.028) (0.029) (0.034)
(0.030)
(20, 40)
-0.024 -0.006 -0.009 -0.001
-0.004
(0.014) (0.014) (0.013) (0.014)
(0.013)
(20, 60)
-0.017 -0.004 -0.006 -0.001
-0.003
(0.010) (0.009) (0.009) (0.009)
(0.009)
(20, 120) -0.008 -0.001 -0.002 -0.000
-0.001
(0.005) (0.004) (0.005) (0.004)
(0.004)
(40, 20)
-0.044 -0.018 -0.021 -0.002
-0.006
(0.021) (0.019) (0.019) (0.023)
(0.021)
(40, 40)
-0.024 -0.007 -0.009 -0.002
-0.004
(0.010) (0.010) (0.010) (0.010)
(0.010)
(40, 60)
-0.015 -0.003 -0.005 -0.001
-0.002
(0.007) (0.007) (0.007) (0.007)
(0.007)
(40, 120) -0.008 -0.001 -0.002 -0.001
-0.001
(0.003) (0.003) (0.003) (0.003)
(0.003)
(60, 20)
-0.044 -0.018 -0.022 -0.002
-0.007
(0.017) (0.016) (0.016) (0.019)
(0.017)
(60, 40)
-0.022 -0.006 -0.008 -0.002
-0.004
(0.009) (0.008) (0.008) (0.008)
(0.008)
(60, 60)
-0.015 -0.003 -0.005 -0.001
-0.003
(0.006) (0.005) (0.005) (0.005)
(0.005)
(60, 120) -0.008 -0.001 -0.002 -0.001
-0.001
(0.003) (0.002) (0.002) (0.002)
(0.002)
(120, 20) -0.044 -0.018 -0.022 -0.002
-0.007
(0.013) (0.011) (0.012) (0.013)
(0.012)
(120, 40) -0.022 -0.006 -0.008 -0.002
-0.004
(0.006) (0.006) (0.006) (0.006)
(0.006)
(120, 60) -0.015 -0.003 -0.005 -0.001
-0.003
(0.004) (0.004) (0.004) (0.004)
(0.004)
(120, 120) -0.008 -0.001 -0.002 -0.001
-0.002
(0.002) (0.002) (0.002) (0.002)
(0.002)
(a) µλ = 0.1, σ 31 = −0.8, σ 21 = −0.4, θ 31 = 0.8, and θ 21 = 0.4.
Table 3: Means Biases and Standard√Deviation of t-statistics
σλ = 1
σ λ = 10
σF = 1
T=20
T=40
T=60
√
σ F = 10
T=20
T=40
T=60
√
σ F = 0.5
T=20
OLS
FMa
FMb
FMc
FMd
OLS
FMa
FMb
FMc
FMd
OLS
F
-1.994
(1.205)
-2.915
(1.202)
-3.465
(1.227)
-1.155
(1.267)
-0.941
(1.101)
-0.709
(1.041)
-1.518
(1.484)
-1.363
(1.248)
-1.158
(1.177)
-0.056
(1.283)
-0.227
(1.054)
-0.195
(0.996)
-0.285
(1.341)
-0.559
(1.141)
-0.574
(1.100)
-0.929
(1.149)
-1.355
(1.127)
-1.552
(1.146)
-0.546
(1.059)
-0.465
(0.913)
-0.308
(0.868)
-0.813
(1.495)
-0.766
(1.207)
-0.568
(1.113)
-0.006
(1.205)
-0.128
(0.912)
-0.074
(0.851)
-0.122
(1.254)
-0.326
(1.049)
-0.261
(1.016)
-2.248
(1.219)
-3.288
(1.221)
-3.926
(1.244)
-1
(1.
-1
(1.
-0
(1.
-1.078
(1.147)
-1.575
(1.131)
-1.809
(1.158)
-0.484
(1.063)
-0.355
(0.917)
-0.155
(0.879)
-0.984
(1.501)
-0.963
(1.214)
-0.776
(1.131)
0.180
(1.220)
0.042
(0.926)
0.111
(0.867)
-0.096
(1.271)
-0.407
(1.063)
-0.390
(1.035)
-0.373
(1.119)
-0.561
(1.097)
-0.588
(1.108)
-0.154
(0.987)
-0.152
(0.844)
-0.041
(0.812)
-0.350
(1.508)
-0.397
(1.179)
-0.247
(1.078)
0.085
(1.194)
-0.014
(0.871)
0.049
(0.811)
-0.006
(1.223)
-0.190
(1.008)
-0.111
(0.983)
-1.427
(1.163)
-2.082
(1.154)
-2.424
(1.192)
-0
(1.
-0
(0.
-0
(0.
-0.054
-0.176
-2.367
(1.217) (1.273) (1.231)
-0.188
-0.385
-3.462
(0.944) (1.087) (1.236)
-0.139
-0.331
-4.149
(0.881) (1.038) (1.249)
the naive CUP- FM.
-1
(1.
-1
(1.
-0
(1.
-2.196
(1.219)
T=40
-3.214
(1.226)
T=60
-3.839
(1.239)
Note: (a) FMa is the
-1.319
-1.606
(1.325) (1.488)
-1.093
-1.415
(1.057) (1.155)
-0.868
-1.217
(1.088) (1.183)
2S-FM, FMb is the
-0.137
-0.327
(1.307) (1.362)
-0.311
-0.576
(1.104) (1.169)
-0.296
-0.602
(1.037) (1.112)
naive 2S-FM, FMc
-1.203
-0.734
-1.008
(1.164) (1.112) (1.488)
-1.752
-0.619
-0.922
(1.148) (0.962) (1.222)
-2.037
-0.446
-0.712
(1.169) (0.908) (1.131)
is the CUP-FM and FMd is
(b) µλ = 0.1, σ 31 = −0.8, σ 21 = −0.4, θ31 = 0.8, and θ21 = 0.4.
Table 4: Means Biases and Standard Deviation
of t-statistics for Different n and T
(n,T)
OLS
FMa
FMb
FMc
FMd
(20, 20)
-1.994 -0.738 -1.032 -0.056
-0.286
(1.205) (1.098) (1.291) (1.283)
(1.341)
(20, 40)
-2.051 -0.465 -0.725 -0.105
-0.332
(1.179) (0.999) (1.126) (1.046)
(1.114)
(20, 60)
-2.129 -0.404 -0.684 -0.162
-0.421
(1.221) (0.963) (1.278) (0.983)
(1.111)
(20, 120) -2.001 -0.213 -0.456 -0.095
-0.327
(1.222) (0.923) (1.083) (0.931)
(1.072)
(40, 20)
-2.759 -1.017 -1.404 -0.103
-0.402
(1.237) (1.116) (1.291) (1.235)
(1.307)
(40, 40)
-2.915 -0.699 -1.075 -0.227
-0.559
(1.202) (1.004) (1.145) (1.054)
(1.141)
(40, 60)
-2.859 -0.486 -0.835 -0.173
-0.493
(1.278) (0.998) (1.171) (1.014)
(1.154)
(40, 120) -2.829 -0.336 -0.642 -0.181
-0.472
(1.209) (0.892) (1.047) (0.899)
(1.037)
(60, 20)
-3.403 -1.252 -1.740 -0.152
-0.534
(1.215) (1.145) (1.279) (1.289)
(1.328)
(60, 40)
-3.496 -0.807 -1.238 -0.255
-0.635
(1.247) (1.016) (1.165) (1.053)
(1.155)
(60, 60)
-3.465 -0.573 -0.987 -0.195
-0.574
(1.227) (0.974) (1.111) (0.996)
(1.100)
(60, 120) -3.515 -0.435 -0.819 -0.243
-0.609
(1.197) (0.908) (1.031) (0.913)
(1.020)
(120, 20) -4.829 -1.758 -2.450 -0.221
-0.760
(1.345) (1.162) (1.327) (1.223)
(1.308)
(120, 40) -4.862 -1.080 -1.679 -0.307
-0.831
(1.254) (1.022) (1.159) (1.059)
(1.143)
(120, 60) -4.901 -0.852 -1.419 -0.329
-0.846
(1.239) (0.964) (1.097) (0.978)
(1.077)
(120, 120) -5.016 -0.622 -1.203 -0.352
-0.908
(1.248) (0.922) (1.059) (0.927)
(1.048)
(a) µλ = 0.1, σ 31 = −0.8, σ 21 = −0.4, θ 31 = 0.8, and θ 21 = 0.4.